# Toward Null State Equations in $d>2$

**Authors:** Kuo-Wei Huang  
**arXiv:** [2308.03229](https://arxiv.org/abs/2308.03229)  
**Date:** 2023-08-06  
**Summary page:** https://www.summarizepaper.com/en/paper/2308.03229v1/

## Key points

- Investigating behavior of heavy-light scalar correlators in four-dimensional conformal field theories (CFTs) with a large central charge
- Focus on multi-stress tensor sector and near-lightcone limit
- Observing that heavy-light scalar correlator obeys a differential equation similar to level-two BPZ equation in two-dimensional CFTs with large central charge
- Equation form: x^3 u_{xxxy} + u = 0 when light scalar has dimension Δ = -1
- Supporting observation through derivation and comparison to holography and conformal bootstrap results
- Exploring cases where Δ takes values -2 and -3, discussing corresponding equations
- Providing solutions to these equations, strengthening observations about heavy-light scalar correlators in four-dimensional CFTs with large central charge
- Contribution to understanding of CFTs by uncovering similarities between two- and four-dimensional theories and providing new insights into heavy-light scalar correlators in higher dimensions
- Deepening knowledge of CFT dynamics with potential implications for theoretical physics.

## Summary

In this paper, the authors investigate the behavior of heavy-light scalar correlators in a class of four-dimensional conformal field theories (CFTs) with a large central charge. They focus on the multi-stress tensor sector of the theory and analyze the correlators in the near-lightcone limit. The authors make an observation that in these four-dimensional CFTs, the heavy-light scalar correlator in the near-lightcone limit obeys a differential equation similar to the level-two BPZ equation found in two-dimensional CFTs with a large central charge. Specifically, they find that when the light scalar has dimension Δ = -1, the correlator satisfies an equation of the form x^3 u_{xxxy} + u = 0. To support their observation, they discuss how this equation can be derived and compare it to previous results obtained via holography and conformal bootstrap. They show that after a suitable change of variables, solutions to this linear partial differential equation in higher dimensions reproduce the near-lightcone correlators obtained through holography and conformal bootstrap. Additionally, they explore other cases where Δ takes values -2 and -3 and discuss their corresponding equations. They provide solutions to these equations as well, further strengthening their observations about heavy-light scalar correlators in four-dimensional CFTs with a large central charge. Overall, this paper contributes to our understanding of CFTs by uncovering similarities between two- and four-dimensional theories and providing new insights into heavy-light scalar correlators in higher dimensions. The findings presented here deepen our knowledge of CFT dynamics and may have implications for various areas of theoretical physics.